Hitting distribution to a quadrant of two-dimensional random walk
type: Item
instance of: scholarly article
title: Hitting distribution to a quadrant of two-dimensional random walk
zbMATH Open document ID: 0964.60057
author: Yasunari Fukai
published in: Kodai Mathematical Journal
publication date: 2001-07-02T00:00:00Z
review text: Let \((S(n))_{n\in\mathbb N_0}\) be a simple random walk on \(\mathbb Z^2\), starting from \(\xi\in\mathbb Z^2\) under \(P_\xi\). Denote by \(\tau_L=\inf\{n\in\mathbb N: S(n)\in L\}\) the hitting time of the third quadrant, \(L=(-\infty,0]^2\), and by \(H_L(\xi,\eta)=P_\xi(S(\tau_L)=\eta)\) the hitting distribution on the boundary of \(L\) when started at \(\xi\notin L\). The Brownian analogues are defined accordingly: For a standard Brownian motion \((W(t))_{t\geq 0}\) on \(\mathbb R^2\), let \(T_L\) be the hitting time of \(L\), and let \(h_L(\xi,\cdot)\) denote the density of the hitting distribution of \(L\), when the motion starts at \(\xi\in\mathbb R^2\setminus L\). The density \(h_L(\xi,\cdot)\) may be explicitly calculated in terms of certain Bessel functions. It is the aim of the paper to derive estimates for the hitting distribution of the simple random walk in terms of the one for the Brownian motion. More precisely, the main result states that, for some absolute constant \(C_0>0\), we have the estimate \[ |H_L(\xi,(-l,0))-h_L(\xi,(-l,0))|\leq C_0[|\xi+(l,0)|^{-3}+|\xi|^{-2/3} l^{-5/3}] \] for any \(\xi\in\mathbb Z^2\setminus L\) and any \(l\in\mathbb N\). Furthermore, it is shown that the powers \(2/3\) and \(5/3\) cannot be improved. Moreover, the existence and positivity of the limits \(\lim_{l\to\infty}l^{5/3}H_L(k,(-l,0))\) and \(\lim_{k\to\infty}k^{2/3}H_L(k,(-l,0))\) for any fixed \(k\in\mathbb N\) resp.\ \(l\in\mathbb N\) are shown. The main tools are estimates for the Green's function of the walk, the reflection principle and an extensive analysis.
reviewed by: W. König
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 1496083
zbMATH Keywords: two-dimensional simple random walk
zbMATH Keywords: hitting distribution
zbMATH Keywords: two-dimensional standard Brownian motion
zbMATH Keywords: Green's function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044155
OpenAlex ID: W2061784781
cites work: Q5753308
cites work: Potential kernel for two-dimensional random walk
cites work: Q4772511
cites work: Hitting probabilities of random walks on \({\mathbb{Z}}^ d\)
cites work: Q3976811
cites work: Q3101597
cites work: Q4131340
DOI: 10.2996/KMJ/1138044155
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description: scientific article; zbMATH DE number 1496083